Simplify the following expression and state the condition under which the simplification is valid. You can assume that $n \neq 0$. $x = \dfrac{5n(3n + 7)}{-3} \div \dfrac{3n(3n + 7)}{5} $
Answer: Dividing by an expression is the same as multiplying by its inverse. $x = \dfrac{5n(3n + 7)}{-3} \times \dfrac{5}{3n(3n + 7)} $ When multiplying fractions, we multiply the numerators and the denominators. $x = \dfrac{ 5n(3n + 7) \times 5 } { -3 \times 3n(3n + 7) } $ $ x = \dfrac{25n(3n + 7)}{-9n(3n + 7)} $ We can cancel the $3n + 7$ so long as $3n + 7 \neq 0$ Therefore $n \neq -\dfrac{7}{3}$ $x = \dfrac{25n \cancel{(3n + 7})}{-9n \cancel{(3n + 7)}} = -\dfrac{25n}{9n} = -\dfrac{25}{9} $